WASHINGTON (CNN) -- The Iraqi military push into the southern city of Basra is not going as well as American officials had hoped, despite President Bush's high praise for the operation, several U.S. officials said Friday.

A burned Iraqi army tank is seen on a street in the southern city of Basra on Friday.

A closely held U.S. military intelligence analysis of the fighting in Basra shows that Iraqi security forces control less than a quarter of the city, according to officials in both the United States and Iraq, and Basra's police units are deeply infiltrated by members of radical Shiite cleric Muqtada al-Sadr's Mehdi Army.

"This is going to go on for a while," one U.S. military official said.

Iraqi forces launched their offensive in Basra this week. Prime Minister Nuri al-Maliki was personally overseeing operations in the southern city against what government officials called "rogue" or "outlaw" militia elements, most loyal to al-Sadr.

During a joint news conference Friday with Australian Prime Minister Kevin Rudd, Bush called the operation "a defining moment in the history of a free Iraq," saying the government is fighting criminals there. Watch more of Bush's comments »

"It was just a matter of time before the government was going to have to deal with it," he said.

The president also hailed the operation as a sign of progress, emphasizing that the decision to mount the offensive was al-Maliki's.

"It was his military planning; it was his causing the troops to go from point A to point B," Bush said. "And it's exactly what a lot of folks here in America were wondering whether or not Iraq would even be able to do it in the first place. And it's happening."

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But since the beginning of the government offensive four days ago, violence also has picked up in a wide area of southern Iraq, including in Baghdad's International Zone -- also known as the Green Zone -- which has been targeted by rocket and mortar attacks.

Coalition bombers have joined in the fight, hitting targets in Basra and Baghdad.

The Basra analysis also shows that militia forces control a wide swath of cities in Iraq's southeast, including areas near the airport, where British forces are located, the officials said.

More than 100 Iraqis have been killed in the fighting, including at least 14 in Baghdad's Sadr City neighborhood. Watch more on Sadr City »

The fighting has sparked fears that a seven-month cease-fire by al-Sadr's Mehdi Army, regarded as a key factor in a dramatic drop in attacks in recent months, could collapse or that the U.S. military will have to bail out the Iraqis.

On Thursday, the Interior Ministry imposed a curfew through the weekend in Baghdad, Hilla, Kut, Diwaniya, Simawa and Basra. Watch more on the curfew »